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Since the S curve is such a basic system evolution trajectory, might it be useful to treat it as an x-axis type, similar to linear and log scales? Is this a thing? An s-scale would transform an underlying linear range (time for eg) by a sigmoid s = 1/(1+e^-t)
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s would initially increase much slower than t, then about the same, the slower again, so equal s-intervals would map to large/small/large t-intervals over range.
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You're plotting the deviation from average, basically, which I think is a thing, just with a mathematical model instead of an actual average.
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The problem is there are too many S-curves. With log scales it doesn't matter much which logarithm you use, it's just a scale factor. Sigmoids have a bunch of parameters.